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Those Fascinating Numbers
Page79(99 of 451)

Those Fascinating Numbers 79 368 • the second solution of σ(n) = 2n + 8; the sequence of numbers satisfying this equation begins as follows: 56, 368, 836, 11 096, 17 816, 45 356, 77 744, 91 388, 128 768, 254 012, 388 076, 2 087 936, 2 291 936, 13 174 976, 29 465 852, 35 021 696,. . . 83 369 • the tenth number k such that k|(10k+1 − 1) (see the number 303). 370 • one of the five numbers which can be written as the sum of the cubes of its digits: 370 = 33 + 73 + 03; the others are 1, 153, 371 and 407. 371 • one of the five numbers which can be written as the sum of the cubes of its digits: 371 = 33 + 73 + 13; the others are 1, 153, 370 and 407. 373 • the largest three digit number (see the number 264) which can be written in two distinct ways as the sum of positive powers of its digits: 373 = 31 + 73 + 33 = 34 + 72 + 35. 376 • the smallest three digit automorphic number: 3762 = 141 376 (see the number 76). 377 • the largest known Fibonacci pseudoprime; the only other one known is 323; • the smallest number n such that the decimal expansion of 2n contains four consecutive zeros (see the number 53). 83It is easy to show that any number n = 2α ·p, where α is a positive integer such that p = 2α+1 −9 is prime, is a solution of σ(n) = 2n + 8: this is the case when α = 3, 4, 8, 10, 16, 20, 32 (and for no other values of α ≤ 100); the solutions corresponding to α = 3, 4, 8, 10 are included in the above list. It is also possible to identify the solutions n of the form 2α · p · q, with p q primes and α a positive integer. This is how one obtains the solutions n = 836, 11 096, 17 816, 77 744, 2 291 936, 13 174 976 and 35 021 696 listed here.